I am looking to find $P(A=0,D=1)$.
We will have something like $(0.5)(.......)$
The dots are the following:
$P(D=1|C=0,B=0)*P(C=0,B=0|A=1) + \dots + P(D=1|C=1,B=1)*P(C=1,B=1|A=1)$
How am I supposed to find $P(C=0,B=0|A=1)$, and is this the correct approach?
Hint: The joint probability factors as follows,
$P_{A,B,C,D} = P_A\cdot P_{B|A}\cdot P_{C|A}\cdot P_{D|B,C}.$
The marginal probability $P_{A,D}$ is computed as follows:
$P_{A,D} (a,d) = \sum_{b\in B}\sum_{c\in C} P_{A,B,C,D}(a,b,c,d).$
Its just a lengthy calculation.
Add: For $P_{B,C|A}(0,0|1)$ you calculate
$$P_{A,B,C}(1,0,0)/P_A(1)$$
where $P_{A,B,C}(a,b,c) = \sum_{d\in D} P_{A,B,C,D}(a,b,c,d)$.